Vertical Shift

To translate the absolute value function f(x)=|x| vertically, you can use the function

g(x)=f(x)+k.

When k>0, the graph of g(x) translated k units up.

When k<0, the graph of g(x) translated k units down.

Horizontal Shift

To translate the absolute value function f(x)=|x| horizontally, you can use the function

g(x)=f(x−h).

When h>0, the graph of f(x) is translated h units to the right to get g(x).

When h<0, the graph of f(x) is translated h units to the left to get g(x).

Stretch and Compression

The stretching or compressing of the absolute value function y=|x| is defined by the function y=a|x| where a is a constant. The graph opens up if a>0 and opens down when a<0.

For absolute value equations multiplied by a constant (for example,y=a|x|),if 0<a<1, then the graph is compressed, and if a>1, it is stretched. Also, if a is negative, then the graph opens downward, instead of upwards as usual.

More generally, the form of the equation for an absolute value function is y=a|x−h|+k. Also:

The vertex of the graph is (h,k).

The domain of the graph is set of all real numbers and the range is y≥k when a>0.

The domain of the graph is set of all real numbers and the range is y≤k when a<0.

The axis of symmetry is x=h.

It opens up if a>0 and opens down if a<0.

The graph y=|x|
can be translated h units horizontally and k units vertically to get the graph of
y=a|x−h|+k.

The graph y=a|x| is wider than the graph of y=|x| if |a|<1 and narrower if |a|>1.